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Unformatted text preview: CHAPTER 9 THE DISCRETE FOURIER TRANSFORM (DFT) In Chapter 7, we learned about the discretetime Fourier transform, which
decomposed discrete time signals into its frequency components. The spectrum
X(Q) is continuous but ﬁnite in length (0, 27:). The discretetime signal could be
inﬁnite in duration. _ When we manipulate signals in digital computers, we use the discrete Fourier
transform because we deal with discretized time signals and discretized spectra. We usually do not deal with inﬁnitely long discretetime signals in digital
computers. We are forced to truncate the signal to some ﬁnite length, N. Truncation of the discretetime signal is accomplished through the use of window
functions, w(n). In Chapter 9, we will learn about discrete Fourier transforms and window
functions. q .‘1 4. 7'.
CHAPTER 9 THE DISCRETE FOURIER TRANSFORM (DFT) The DF T equations are derived from the DTFT equations by considering the spectrum X(Q) to be sampled function. X(Q) X622) ' Recall the DTFT: X(o) = Elﬁn)?”
If we restrict the spectrum to discrete values of X(Q), the transform becomes: X(Qk) = ﬂamers“ HEUJ X(kAQ)'= impM" Substituting A9 = 2Einto the above , gives : N j27dﬂ1 Xﬂc) = ix(n)e N =4: CHAPTER 9 THE DISCRETE F OURIER TRANSFORM (DFT) Up to this point, the discretetime signal is still inﬁnite in length. However, from a
practical point of View, we are restricted to ﬁnite length signals in digital computers. To
achieve the ﬁnite length discrete signal, we multiply the inﬁnite length signal by a
window function which truncates the signal. The simplest window function is the
rectangle, w(n): {1 OSnSN—l
W(n)= w(n) 0 Otherwise You will be working with this as well as Hamming and Hanning windows in
Lab this coming week. Multiplying by this window ﬁJnction does not effect the amplitude of the discrete signal,
it simply has the effect of shortening the length to N. With this window function applied to a discretetime signal, the DF T becomes: —}'2:mk X(k)=§x(n)e N 93 6M CHAPTER 9 THE DISCRETE FOURIER TRANSFORM (DFT) The inverse DFT is: 1 N—l 13—73“!
x(n)— ' N —§X(k)eT
The spectrum, as determined by the DFT, is periodic with period N:
N—l —j2:m(k+N) Nu] —j2mk e'—JZmN —j2mk j2;mk
X(k + N) = Zx(n)e N = Zx(n)e N  —zx(n)e” e“”"— — Zx(n)e” X(k + N) : X(k) Similarly, the time signal, determined from the spectrum, is periodic with period N.
The proof is similar to that above: The DFT is not an efficient way to compute the Fourier transform on a digital
computer. It takes N*N complex operations to compute, N is the length of the
discretetime signal so for N = 16, there are 256 complex operations required. The Fast Fourier Transform is an efficientway to implement the DFT on a digital
computer. It reduces the number of operations to Nlog2N. The table below
compares the complex operations for the DF T and FFT. 65536 2048 1024 1048576 10240 CHAPTER 9 THE DISCRETE FOURIER TRANSFORM (DFT) We can gain some insight into why the FFT by examining the transform for N = 8: _j2ank X(k) = ix(n)e 3 g 12d: _ 1‘2:er _ j21r3k _j27r4k _ 1255i: _ 121:5}: Jam
3 X(k):x(0)+x(l)e 3 +x(2)e 3 +x(3)e 3 +x(4)e 3 +x(5)e 3 +x(6)e 3 +x(7)e Next group the even terms and the odd terms : AszZk _j2:r4k _j2r:6k _j2a:k _j2:r3k _j2:r5k _j27r7k
s X(k):x(0)+x(2)e 3 +x(4)e 3 +x(6)e 3 +x(1)e 3 +x(3)e 3 +x(5)e 3 +x(7)e Next within each of these groups, group the even terms and the odd terms :
_j2£4k _j2z2k ijzﬁk _j2nk _j2z5k ~j27r3k _j2n7k X(k)=x(0)+x(4)e 3 +x(2)e 3 +x(6)e 3 +x(l)e 3 +x(5)e 3 +x(3)e 3 +x(7)e 3 We can write this as :
_j21r4k ijn'Zk Jam: _j27d't _j2n'4k _j2:r3k ﬁj21r4k X(k) = (x(O) + x(4)e 3 )+ e 3 (x(2) + x(6)e 3 )+ e T(x(1) + x(5)e 3 )+ e 3 (x(3) + x(7)e 3 )
Each of the groups of terms in parentheses is a two point DFT :
Let k = 0 g 12mm _ 12320 _ 12;:40 _j27r0 _j27r40 _ 12;:30 _j21r40 'X(0) = (x(O) + x(4)e 3 )+ e 3 (x(Z) + x(6)e 3 )+ e 3 (x(1)+ x(5)e 3 )+ e 3 (x(3) + x(7)e 3 ) _j2ﬂ0 Lete 3 2 W30 =1
X(0) = (x(O) + x(4)W8° ) + W30 (x(z) + x(6)W8°) + W80(x(1) + x(5)W80) + W30 (x(3) + x(7)W80) 0, é
CHAPTER 9 THE DISCRETE FOURIER TRANSFORM (DFT) This is ﬂow graph of the X(0) component of the FFT of x(n).
Compare this with the ﬁgure 9.5.5 on page 432 of the text. x(0) + x(4)W;’ W) + x(4)W8°) + (x0) + x<6)Ws°)
+(x(1)+x(5)W8°)+(x(3)+x(7)Ws°) (36(1) + x(5)WsO) + (16(3) + x(7)Wsu) x(3) + x(7)W X(3) M W x(7) q 7
CHAPTER 9 THE DISCRETE FOURIER TRANSFORM (DFT) Let k = 2
_j211'42 _j21r22 _j21'r42 _j21'rZ _j2:r42 _j2£32 _j21r42
'X(k) : (x(0)+x(4)e 3 )+e 3 (x(2)+x(6)e 3 )+e 3 (x(1)+x(5)e 8 )+e 3 (x(3)+x(7)e 3 )
_j27r42 _j27r8 Lete 8 =e 3 =e‘j2”=l=W30=1 _j21r22 32:4
9 8 = e 3 2 W34 _EE
3 3 2 W32 _j2n'32 _j2:r6
e 3 = e 3 2 WE6 X(2) = (x(0) + x(4)W8°) + W8“ (x(2) + x(6)Ws° ) + W32(x(1) + x(5)W30) + W86 (x(3) + x(7)W80)
Notice that the terms in parentheses are the same terms in X(0), so they do not have to be recalculated.
They simply have to be retrieved from memory. We can also write X(2) in the following form : X(2) = mm + scam") + (x(2) + x(6)Ws°)W;‘ +[(x(1) + x(5)Wg° ) + (x6) + x<7>W8°)W341W: 6).?
CHAPTER 9 THE DISCRETE FOURIER TRANSFORM (DFT) This is ﬂow graph of the X(0) component of the FFT of x(n).
Compare this with the ﬁgure 9.5.5 on page 432 of the text. x(0) + x(4)W80 me) + x(4)Ws°) + W) + Men/3W:
+ [(241) + x(5)Ws°) + (x6) + x(7)Ws°)Ws‘ 1W: _—[email protected] X(2) W
/ (“owner/re“)
x(6) +(x(2)+x(6)Ws°)Ws‘ x(1) + x(5)W8°
x(1) o7 :) W32
W0
x(5) 6
x(3) + x(7)W8° <x(1)+x(5)Wa°>
W” +(x<3)+x<7)Wg°)Ws“ X(7) ...
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 Spring '10
 Rajasekaran

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